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Write the equation of the parabola that passes through the points $(- 2,0)$ , $(1, - 4)$ , and $(3,0)$ . Write your answer in the form $y=a(x-p)(x-q)$ , where $a$ , $p$ , and $q$ are integers, decimals, or simplified fractions.

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Write a quadratic function from its x-intercepts and another point

Full solution

Q. Write the equation of the parabola that passes through the points

(- 2,0)

,

(1, - 4)

, and

(3,0)

. Write your answer in the form

y=a(x-p)(x-q)

, where

a

,

p

, and

q

are integers, decimals, or simplified fractions.

Identify x-intercepts: Identify the x-intercepts from the given points $(-2,0)$ and $(3,0)$ . These points indicate where the parabola crosses the x-axis, so $p = -2$ and $q = 3$ .
Formulate parabola equation: Formulate the equation of the parabola using the identified $x$ -intercepts. The general form is $y = a(x - p)(x - q)$ . Substituting $p$ and $q$ , we get $y = a(x + 2)(x - 3)$ .
Find value of $a$ : Use the point $(1, –4)$ to find the value of $a$ . Substitute $x = 1$ and $y = -4$ into the equation: $-4 = a(1 + 2)(1 - 3)$ .
Solve for $a$ : Simplify and solve for $a$ : $-4 = a(3)(-2)$ , $-4 = -6a$ , $a = -4 / -6 = 2/3$ .

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